December  2012, 2(4): 331-359. doi: 10.3934/mcrf.2012.2.331

Carleman inequalities for the two-dimensional heat equation in singular domains

1. 

Faculté de Mathématiques, laboratoire AMNEDP, U.S.T.H.B, B.P. 32, El-Alia, 16111 ALGER, Algeria, Algeria, Algeria, Algeria

Received  January 2012 Revised  May 2012 Published  October 2012

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we prove a Carleman inequality and we deduce a result of controllability.
Citation: Abdelhakim Belghazi, Ferroudja Smadhi, Nawel Zaidi, Ouahiba Zair. Carleman inequalities for the two-dimensional heat equation in singular domains. Mathematical Control & Related Fields, 2012, 2 (4) : 331-359. doi: 10.3934/mcrf.2012.2.331
References:
[1]

A-H. Belghazi, Null controllability of three-dimensional heat equation for singular domains and stratified media spectral inequality,, in preparation., ().   Google Scholar

[2]

A. H. Belghazi, F. Smadhi, N. Zaidi and O. Zair, Carleman inequalities for the heat equation in singular domains,, C. R. Acad. Sci. Paris, 348 (2010), 277.  doi: 10.1016/j.crma.2010.02.007.  Google Scholar

[3]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, On the controllability of linear parabolic equations with an arbitrary control location for stratified media,, C. R. Acad. Sci. Paris, 344 (2007), 357.   Google Scholar

[4]

R. Bey, J. P Lohéac and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation,, J. Math. Pures Appl., 78 (1999), 1043.   Google Scholar

[5]

L. Bourgeois, A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners,, C. R. Acad. Sci. Paris, 345 (2007).   Google Scholar

[6]

H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).   Google Scholar

[7]

N. Burq, Contrôle de l'équation des ondes dans des ouverts contenants des coins,, Bull. Soc. math. Franc, 126 (1998), 601.   Google Scholar

[8]

T. Carleman, Sur un problème d'unicit'e pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes,, Ark. Mat. Astr. Fys., 26 (1939), 1.   Google Scholar

[9]

A. Doubova, A. Osses and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients,, ESAIM Control Optim. Calc. Var., 8 (2002), 621.   Google Scholar

[10]

H-O. Fattorini and D-L.Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rational Mech. Anal., 43 (1971), 272.   Google Scholar

[11]

H-O. Fattorini and D-L.Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (): 45.   Google Scholar

[12]

E. Fernandez-Cara and S. Guerrero, Global Carleman Inequalities For Parabolic Systems And application To Controllability,, SIAM J. Control Optim, 45 (2006), 1395.  doi: 10.1137/S0363012904439696.  Google Scholar

[13]

A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations,, Lecture Notes Series 34, (1996).   Google Scholar

[14]

P. Grisvard, Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités,, Journal Maths. Pures et Appl., 68 (1989), 215.   Google Scholar

[15]

P. Grisvard, "Singularities in Boundary Value Problems,", Springer-Verlag, (1992).   Google Scholar

[16]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Pitman, (1985).   Google Scholar

[17]

A. Heibig and M. A. Moussaoui, Exact controllability of the wave equation for domains with slits and for mixed boundary conditions,, Discrete and continuous dynamical systems, 2 (1996), 367.   Google Scholar

[18]

L. Hörmander, "Linear Partial Differential Operators,", Springer-verlag, (1963).   Google Scholar

[19]

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients,, J. Differential Equations, 233 (2007), 417.  doi: 10.1016/j.jde.2006.10.005.  Google Scholar

[20]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential. Equations, 20 (1995), 335.   Google Scholar

[21]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, to appear in ESAIM Control Optim. Calc. Var., ().   Google Scholar

[22]

M. A. Moussaoui and B. K. Saadallah, Régularité des coefficients de propagation de singularités de l'équation de la chaleur dans un domaine polygonal plan,, C. R. Acad. Sci. Paris, 293 (1981), 297.   Google Scholar

[23]

S. Nicaise, Exact controllability of a pluridimensional coupled problem,, Rev. Math. Univ. Complut. Madrid, 5 (1992), 91.   Google Scholar

[24]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983).   Google Scholar

[25]

J. Zabczyk, "Mathematical Control Theory: An Introduction,", Birkhäuser, (1995).   Google Scholar

show all references

References:
[1]

A-H. Belghazi, Null controllability of three-dimensional heat equation for singular domains and stratified media spectral inequality,, in preparation., ().   Google Scholar

[2]

A. H. Belghazi, F. Smadhi, N. Zaidi and O. Zair, Carleman inequalities for the heat equation in singular domains,, C. R. Acad. Sci. Paris, 348 (2010), 277.  doi: 10.1016/j.crma.2010.02.007.  Google Scholar

[3]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, On the controllability of linear parabolic equations with an arbitrary control location for stratified media,, C. R. Acad. Sci. Paris, 344 (2007), 357.   Google Scholar

[4]

R. Bey, J. P Lohéac and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation,, J. Math. Pures Appl., 78 (1999), 1043.   Google Scholar

[5]

L. Bourgeois, A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners,, C. R. Acad. Sci. Paris, 345 (2007).   Google Scholar

[6]

H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).   Google Scholar

[7]

N. Burq, Contrôle de l'équation des ondes dans des ouverts contenants des coins,, Bull. Soc. math. Franc, 126 (1998), 601.   Google Scholar

[8]

T. Carleman, Sur un problème d'unicit'e pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes,, Ark. Mat. Astr. Fys., 26 (1939), 1.   Google Scholar

[9]

A. Doubova, A. Osses and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients,, ESAIM Control Optim. Calc. Var., 8 (2002), 621.   Google Scholar

[10]

H-O. Fattorini and D-L.Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rational Mech. Anal., 43 (1971), 272.   Google Scholar

[11]

H-O. Fattorini and D-L.Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (): 45.   Google Scholar

[12]

E. Fernandez-Cara and S. Guerrero, Global Carleman Inequalities For Parabolic Systems And application To Controllability,, SIAM J. Control Optim, 45 (2006), 1395.  doi: 10.1137/S0363012904439696.  Google Scholar

[13]

A. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations,, Lecture Notes Series 34, (1996).   Google Scholar

[14]

P. Grisvard, Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités,, Journal Maths. Pures et Appl., 68 (1989), 215.   Google Scholar

[15]

P. Grisvard, "Singularities in Boundary Value Problems,", Springer-Verlag, (1992).   Google Scholar

[16]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Pitman, (1985).   Google Scholar

[17]

A. Heibig and M. A. Moussaoui, Exact controllability of the wave equation for domains with slits and for mixed boundary conditions,, Discrete and continuous dynamical systems, 2 (1996), 367.   Google Scholar

[18]

L. Hörmander, "Linear Partial Differential Operators,", Springer-verlag, (1963).   Google Scholar

[19]

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients,, J. Differential Equations, 233 (2007), 417.  doi: 10.1016/j.jde.2006.10.005.  Google Scholar

[20]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential. Equations, 20 (1995), 335.   Google Scholar

[21]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, to appear in ESAIM Control Optim. Calc. Var., ().   Google Scholar

[22]

M. A. Moussaoui and B. K. Saadallah, Régularité des coefficients de propagation de singularités de l'équation de la chaleur dans un domaine polygonal plan,, C. R. Acad. Sci. Paris, 293 (1981), 297.   Google Scholar

[23]

S. Nicaise, Exact controllability of a pluridimensional coupled problem,, Rev. Math. Univ. Complut. Madrid, 5 (1992), 91.   Google Scholar

[24]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983).   Google Scholar

[25]

J. Zabczyk, "Mathematical Control Theory: An Introduction,", Birkhäuser, (1995).   Google Scholar

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